As detailed in the previous sections, OT allows one to compute assignments (or permutations) accounting for a metric similarity \( between samples.However such metric prior is not straightforwardly available in some tasks, for instance if the measures are defined on two different spaces \)\( and \)\( and no cost \)  is defined on \(\times \) .
Such a setting arises with graph data [1]}.
They embed objects such as social networks [2]}, molecules in biochemistry [3]} or shapes [4]}.
One example where we seek an assignment between two graphs is the graph isomorphism problem, which is known to be NP-hard [5]}.
Another practical setting where no cost is available happens when we seek an assignment between measures defined on Euclidean spaces of different dimensions, as is the case in biology when measurements are performed under distinct modalities [6]}.
